Distributed Solution of GNEP over Networks via the Douglas-Rachford Splitting Method

Abstract

We consider a generalized Nash equilibrium problem (GNEP) for a network of players. Each player tries to minimize a local objective function subject to some resource constraints where both the objective functions and the resource constraints depend on other players' decisions. By conducting equivalent transformations on the local optimization problems and introducing network Lagrangian, we recast the GNEP into an operator zero-finding problem. An algorithm is proposed based on the Douglas-Rachford method to distributedly find a solution. The proposed algorithm requires milder conditions compared to the existing methods. We prove the convergence of the proposed algorithm to an exact variational generalized Nash equilibrium under two different sets of assumptions. Our algorithm is validated numerically through the example of a Nash-Cournot production game.

Nash equilibrium seeking under partial decision information: Monotonicity, smoothness and proximal-point algorithms

“…3] cannot. Our examples also show a significant gap between C4 and the condition C3, employed e.g., in [18,Th. 3].…”

“…Remark 1: In [17] we have proven (linear) convergence of Algorithm 1 assuming C6; under the weaker C4, Theorem 1 leverages the general results for the proximal-point algorithm of restricted (merely) monotone games [16]. With respect to [16] and to the Douglas-Rachford algorithm in [18], we use a different limiting argument in our proof, which does not require F to be Lipschitz continuous (or even continuous). The core idea is to show that the operator J Φ −1 Aα is continuous, even if A α is not (nor is maximally monotone).…”

“…For instance, Theorem 1 can be applied to the games in Examples 2 to 4, while [16,Th. 2], [18,Th. 3] cannot.…”

Nash equilibrium seeking under partial-decision information: Monotonicity, smoothness and proximal-point algorithms

“…Under the weaker C4, Theorem 1 leverages the general results for the proximal-point algorithm of restricted (merely) monotone games [16]. With respect to [16] and to the Douglas-Rachford algorithm in [18], we use a different limiting argument in our proof, which does not require F to be Lipschitz continuous (or even continuous). The core idea is to show that the operator J Φ −1 Aα is continuous, even if A α might not (nor is maximally monotone).…”

Continuous-time fully distributed generalized Nash equilibrium seeking for multi-integrator agents

“…To overcome this complication, vanishing step sizes can be used to drive the error to zero [11], at the price of slow convergence. Alternatively, several fixed-step algorithms for GNE seeking were derived based on operator-theoretic methods and on the use of preconditioning [8], [9], [12], [7]. Unfortunately, this approach comes with important limitations, such as extending the analysis to time-varying setups.…”